Hands-on Exercise 2B

Author

yang yayong

Published

August 27, 2024

Modified

September 3, 2024

5  2nd Order Spatial Point Patterns Analysis Methods

5.1 Overview

Spatial Point Pattern Analysis is the evaluation of the pattern or distribution, of a set of points on a surface. The point can be location of:

  • events such as crime, traffic accident and disease onset, or

  • business services (coffee and fastfood outlets) or facilities such as childcare and eldercare.

Using appropriate functions of spatstat, this hands-on exercise aims to discover the spatial point processes of childecare centres in Singapore.

5.2 The data

To provide answers to the questions above, three data sets will be used. They are:

  • CHILDCARE, a point feature data providing both location and attribute information of childcare centres. It was downloaded from Data.gov.sg and is in geojson format.

  • MP14_SUBZONE_WEB_PL, a polygon feature data providing information of URA 2014 Master Plan Planning Subzone boundary data. It is in ESRI shapefile format. This data set was also downloaded from Data.gov.sg.

  • CostalOutline, a polygon feature data showing the national boundary of Singapore. It is provided by SLA and is in ESRI shapefile format.

5.3 Installing and Loading the R packages

pacman::p_load(sf, raster, spatstat, tmap, tidyverse)

5.4 Spatial Data Wrangling

5.4.1 Importing the spatial data

In this section, st_read() of sf package will be used to import these three geospatial data sets into R.

childcare_sf <- st_read("data/ChildCareServices.geojson") %>%
  st_transform(crs = 3414)
Reading layer `ChildCareServices' from data source 
  `/Users/yangyayong/Downloads/学校文件/smu文件/Term 3/G/yyyirene/ISSS626-GAA/Hands-on_Ex/Hands-on_Ex02/data/ChildCareServices.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1925 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6878 ymin: 1.247759 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84
sg_sf <- st_read(dsn = "data", layer="CostalOutline")
Reading layer `CostalOutline' from data source 
  `/Users/yangyayong/Downloads/学校文件/smu文件/Term 3/G/yyyirene/ISSS626-GAA/Hands-on_Ex/Hands-on_Ex02/data' 
  using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21
mpsz_sf <- st_read(dsn = "data", 
                layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `/Users/yangyayong/Downloads/学校文件/smu文件/Term 3/G/yyyirene/ISSS626-GAA/Hands-on_Ex/Hands-on_Ex02/data' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

5.4.2 Mapping the geospatial data sets

tmap_mode('view')
tmap mode set to interactive viewing
tm_shape(childcare_sf)+
  tm_dots()

5.5 Geospatial Data wrangling

Although simple feature data frame is gaining popularity again sp’s Spatial* classes, there are, however, many geospatial analysis packages require the input geospatial data in sp’s Spatial* classes. In this section, you will learn how to convert simple feature data frame to sp’s Spatial* class.

5.5.1 Converting from sf format into spatstat’s ppp format

Now, we will use as.ppp() function of spatstat to convert the spatial data into spatstat’s ppp object format

childcare_ppp <- as.ppp(childcare_sf)
Warning in as.ppp.sf(childcare_sf): only first attribute column is used for
marks
childcare_ppp
Marked planar point pattern: 1925 points
marks are of storage type  'character'
window: rectangle = [11810.03, 45404.24] x [25596.33, 49300.88] units
plot(childcare_ppp)
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 1925 symbols are shown in the symbol map

plot(childcare_ppp)
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 1925 symbols are shown in the symbol map

summary(childcare_ppp)
Marked planar point pattern:  1925 points
Average intensity 2.417323e-06 points per square unit

Coordinates are given to 11 decimal places

marks are of type 'character'
Summary:
   Length     Class      Mode 
     1925 character character 

Window: rectangle = [11810.03, 45404.24] x [25596.33, 49300.88] units
                    (33590 x 23700 units)
Window area = 796335000 square units

5.5.2 Handling duplicated points

any(duplicated(childcare_ppp))
[1] FALSE
multiplicity(childcare_ppp)
   [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
  [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
  [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [112] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [149] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [186] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [223] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [260] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [297] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [334] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [371] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [408] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [445] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [482] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [519] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [556] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [593] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [630] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [667] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [704] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [741] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [778] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [815] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [852] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [889] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [926] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [963] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1000] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1037] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1074] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1111] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1148] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1185] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1222] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1259] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1296] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1333] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1370] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1407] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1444] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1481] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1518] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1555] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1592] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1629] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1666] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1703] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1740] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1777] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1814] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1851] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1888] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
[1925] 1
tmap_mode('view')
tmap mode set to interactive viewing
tm_shape(childcare_sf) +
  tm_dots(alpha=0.4, 
          size=0.05)
tmap_mode('plot')
tmap mode set to plotting
childcare_ppp_jit <- rjitter(childcare_ppp, 
                             retry=TRUE, 
                             nsim=1, 
                             drop=TRUE)
any(duplicated(childcare_ppp_jit))
[1] FALSE

5.5.3 Creating owin object

sg_owin <- as.owin(sg_sf)
plot(sg_owin)

summary(sg_owin)
Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

5.5.4 Combining point events object and owin object

childcareSG_ppp = childcare_ppp[sg_owin]
summary(childcareSG_ppp)
Marked planar point pattern:  1925 points
Average intensity 2.653796e-06 points per square unit

Coordinates are given to 11 decimal places

marks are of type 'character'
Summary:
   Length     Class      Mode 
     1925 character character 

Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

5.5.4.1 Extracting study area

pg <- mpsz_sf %>%
  filter(PLN_AREA_N == "PUNGGOL")
tm <- mpsz_sf %>%
  filter(PLN_AREA_N == "TAMPINES")
ck <- mpsz_sf %>%
  filter(PLN_AREA_N == "CHOA CHU KANG")
jw <- mpsz_sf %>%
  filter(PLN_AREA_N == "JURONG WEST")
par(mfrow=c(2,2))
plot(pg, main = "Ponggol")
Warning: plotting the first 9 out of 15 attributes; use max.plot = 15 to plot
all

plot(tm, main = "Tampines")
Warning: plotting the first 9 out of 15 attributes; use max.plot = 15 to plot
all

plot(ck, main = "Choa Chu Kang")
Warning: plotting the first 10 out of 15 attributes; use max.plot = 15 to plot
all

plot(jw, main = "Jurong West")
Warning: plotting the first 9 out of 15 attributes; use max.plot = 15 to plot
all

5.5.4.2 Converting sf objects into owin objects

pg_owin = as.owin(pg)
tm_owin = as.owin(tm)
ck_owin = as.owin(ck)
jw_owin = as.owin(jw)

5.5.4.3 Combining childcare points and the study area

childcare_pg_ppp = childcare_ppp_jit[pg_owin]
childcare_tm_ppp = childcare_ppp_jit[tm_owin]
childcare_ck_ppp = childcare_ppp_jit[ck_owin]
childcare_jw_ppp = childcare_ppp_jit[jw_owin]
childcare_pg_ppp.km = rescale(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale(childcare_jw_ppp, 1000, "km")
par(mfrow=c(2,2))
plot(childcare_pg_ppp.km, main="Punggol")
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 72 symbols are shown in the symbol map
plot(childcare_tm_ppp.km, main="Tampines")
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 117 symbols are shown in the symbol map
plot(childcare_ck_ppp.km, main="Choa Chu Kang")
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 74 symbols are shown in the symbol map
plot(childcare_jw_ppp.km, main="Jurong West")
Warning in default.charmap(ntypes, chars): Too many types to display every type
as a different character
Warning: Only 10 out of 109 symbols are shown in the symbol map

5.6 Second-order Spatial Point Patterns Analysis

5.7 Analysing Spatial Point Process Using G-Function

The G function measures the distribution of the distances from an arbitrary event to its nearest event. In this section, you will learn how to compute G-function estimation by using Gest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

5.7.1 Choa Chu Kang planning area

5.7.1.1 Computing G-function estimation

G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK, xlim=c(0,500))

This figure shows that the nearest neighbor distance between nurseries may have a clustering effect in a small range (that is, nurseries may be concentrated in certain areas), while the distribution of nurseries tends to be random in a larger distance range.

The black solid line is higher than the red dashed line at a given r value, which indicates that the nurseries have more nearest neighbors in these distance ranges, which means that these nurseries may tend to cluster at these distances.

5.7.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

Monte Carlo test with G-fucntion

G_CK.csr <- envelope(childcare_ck_ppp, Gest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_CK.csr)

Since the black solid line G_obs(r) is basically within the gray shaded area (i.e. the simulated confidence interval), we cannot reject the null hypothesis that the distribution of nurseries in Choa Chu Kang area can be considered as a random distribution in statistics.

5.7.2 Tampines planning area

5.7.2.1 Computing G-function estimation

G_tm = Gest(childcare_tm_ppp, correction = "best")
plot(G_tm)

The figure shows that the distribution of childcare centers in the Tampines region shows clear clustering. As the distance r increases, the nearest neighbor distance between nurseries becomes shorter, and the probability of finding the nearest neighbor nursery is significantly higher than in the case of a random distribution.

Therefore, we can consider that the distribution of nurseries in the Tampines area is not random, but has a certain concentration tendency, possibly due to the concentration of these nurseries in certain areas

5.7.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

G_tm.csr <- envelope(childcare_tm_ppp, Gest, correction = "all", nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_tm.csr)

Since the actual G-function (solid black line) exceeds the confidence interval of the random distribution (gray area), this means that the distribution of nurseries in the Tampines region is not random but has a significant tendency to cluster.

Specifically, within a range of less than 200 meters, the distance between nurseries is shorter and the concentration is higher, indicating that these nurseries tend to be geographically spatially clustered in certain areas.

5.8 Analysing Spatial Point Process Using F-Function

The F function estimates the empty space function F(r) or its hazard rate h(r) from a point pattern in a window of arbitrary shape. In this section, you will learn how to compute F-function estimation by using Fest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

5.8.1 Choa Chu Kang planning area

5.8.1.1 Computing F-function estimation

The code chunk below is used to compute F-function using Fest() of spatat package.

F_CK = Fest(childcare_ck_ppp)
plot(F_CK)

This F-function analysis shows that the spatial distribution pattern of nurseries in the Choa Chu Kang area is close to random distribution. The F-function results obtained by different correction methods also show that the distribution of these nurseries does not show significant spatial clustering or sparseness.

5.8.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

Monte Carlo test with F-fucntion

F_CK.csr <- envelope(childcare_ck_ppp, Fest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_CK.csr)

This indicates that the distribution of nurseries in the Choа Chu Kang area is not significantly different from a completely random distribution. The actual observed nursery distribution is very close to the random model, falling within the confidence interval of the random distribution generated by Monte Carlo simulations

5.8.3 Tampines planning area

5.8.3.1 Computing F-function estimation

F_tm = Fest(childcare_tm_ppp, correction = "best")
plot(F_tm)

5.8.3.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

F_tm.csr <- envelope(childcare_tm_ppp, Fest, correction = "all", nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_tm.csr)

Hypothesis test: Because the actual F-function is much lower than the theoretical F-function and the confidence interval of the random distribution, we can reject the null hypothesis that the distribution of childcare centers in the Tampines area is not random.

Final explanation: The distribution of childcare centers in the Tampines area shows a high spatial dispersion rather than a random distribution. This may mean that the childcare centers are more geographically dispersed with larger empty areas, or the distribution of these childcare centers is affected by some factors, resulting in relatively large distances between them.

5.9 Analysing Spatial Point Process Using K-Function

K-function measures the number of events found up to a given distance of any particular event. In this section, you will learn how to compute K-function estimates by using Kest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

5.9.1 Choa Chu Kang planning area

5.9.1.1 Computing K-fucntion estimate

K_ck = Kest(childcare_ck_ppp, correction = "Ripley")
plot(K_ck, . -r ~ r, ylab= "K(d)-r", xlab = "d(m)")

5.9.1.2 Performing Complete Spatial Randomness Test

K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")

Hypothesis testing: Since the actual observed K-function mainly falls within the confidence interval of the random distribution of the Monte Carlo simulation, it shows that we cannot reject the null hypothesis (the nursery distribution is random). Although the solid line is slightly higher than the theoretical value, the difference is not significant and therefore does not support the conclusion that there is significant clustering in nurseries.

Final note: The distribution of nurseries in the Choa Chu Kang area shows a near-random distribution pattern overall, with a slight tendency to cluster, but this trend is not significant enough to deviate significantly from a completely random distribution.

5.9.2 Tampines planning area

5.9.2.1 Computing K-fucntion estimation

K_tm = Kest(childcare_tm_ppp, correction = "Ripley")
plot(K_tm, . -r ~ r, 
     ylab= "K(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_tm.csr, . - r ~ r, 
     xlab="d", ylab="K(d)-r", xlim=c(0,500))

5.10 Analysing Spatial Point Process Using L-Function

In this section, you will learn how to compute L-function estimation by using Lest() of spatstat package. You will also learn how to perform monta carlo simulation test using envelope() of spatstat package.

5.10.1 Choa Chu Kang planning area

5.10.1.1 Computing L Fucntion estimation

L_ck = Lest(childcare_ck_ppp, correction = "Ripley")
plot(L_ck, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)")

5.10.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value if smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")

In the figure you provided, the black solid line deviates slightly from the red dotted line in some intervals, but most of the area is within the gray shaded area. This implies that within these intervals, the distribution of nurseries does not show a clear clustering or uniform pattern. Therefore, based on this result, the null hypothesis may not be significantly rejected, indicating that the distribution pattern of nurseries in the Choa Chu Kang area may be close to a random distribution.

5.10.2 Tampines planning area

5.10.2.1 Computing L-fucntion estimate

L_tm = Lest(childcare_tm_ppp, correction = "Ripley")
plot(L_tm, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

5.10.2.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

The code chunk below will be used to perform the hypothesis testing.

L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_tm.csr, . - r ~ r, 
     xlab="d", ylab="L(d)-r", xlim=c(0,500))

The solid line exceeds the gray shaded area: The black solid line clearly exceeds the gray shaded area, especially in areas with larger distances (>100). This means that the observed nursery distribution deviates significantly from a random distribution pattern.

Impact on the null hypothesis: According to the results in the figure, since the solid line significantly exceeds the gray shaded area, this indicates that the distribution of childcare centers in the Tampines area is not random. Therefore, we can reject the null hypothesis (Ho) and accept the alternative hypothesis (H1), that is, the distribution of nurseries in Tampines has significant clustering or other non-random characteristics.